Deformation to the normal cone and specialization

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth normal-sequence suppliers. For a closed immersion $i:X\hookrightarrow Y$ with ideal $\mathcal{I}$, put $C_{X}Y={\mathrm{Spec}}_X({⨁}_{n\ge 0}{\mathcal{I}}^n\mathrm{/}{\mathcal{I}}^{n+1})$ and $M={\mathrm{Bl}}_{X\times \{\mathrm{\infty }\}}(Y\times {\mathbb{P}}^1)$. Let $B={\mathrm{Bl}}_{X}Y$ be the strict transform of $Y\times \{\mathrm{\infty }\}$. The fibre at infinity is the sum $M_{\mathrm{\infty }}=B+\mathbb{P}(C_{X}Y\oplus 1)$ of effective Cartier divisors, whose intersection is $\mathbb{P}(C_{X}Y)$; this is a union with a common boundary, not a disjoint union. Projectivized cones here use the lines convention $\mathrm{Proj}({\mathrm{gr}}_{\mathcal{I}}{\mathcal{O}}_Y[S])$. Off infinity $M\cong Y\times {\mathbb{A}}^1$, and $D_i:=M\setminus B$ has special fibre $C_{X}Y$ and ordinary fibres $Y$. It is flat over ${\mathbb{P}}^1$. The specialization ${\sigma }_i:A_m(Y)\to A_m(C_{X}Y)$ is obtained by extending the flat pullback of a cycle to $D_i$, then taking Cartier Gysin at infinity. It sends an integral $[V]$ to the fundamental cycle $[C_{X\cap V}V]$ pushed to $C_{X}Y$. For a regular immersion of codimension $d$, the normal cone equals the rank-$d$ normal bundle $N_{X\mathrm{/}Y}$. The construction works for finite type schemes over a field, with locally finite cycles in the locally finite type case.

Well-definedness. The charts of $M$ near infinity are computed directly: over an affine open $\mathrm{Spec}A\subseteq Y$ with $X=\mathrm{Spec}(A\mathrm{/}I)$ and $t$ the coordinate vanishing at infinity, the $t$-chart of the blowup is the affine blowup algebra $A[t,I\mathrm{/}t]={\sum }_{n\ge 0}{t}^{-n}{I}^n[t]\subseteq A[t,t^{-1}]$, and multiplication by $t$ is injective with quotient ${\mathrm{gr}}_{I}A$; in the chart of a generator $a\in I$ the ring is $A[I\mathrm{/}a][t\mathrm{/}a]$, and $t=a(t\mathrm{/}a)$ exhibits the fibre at $t=0$ as the sum of the exceptional divisor $a=0$ and the strict transform $t\mathrm{/}a=0$, whose charts are ${\mathrm{Bl}}_{X}Y$ and whose intersection is $\mathrm{Proj}{\mathrm{gr}}_{I}A$. This identifies the fibre at infinity as the stated union of effective Cartier divisors, without assuming $Y$ integral or the centre Cartier, and shows that, away from infinity, $M\setminus B\cong Y\times {\mathbb{A}}^1$, while its special fibre is $C_{X}Y$; the charts are torsion-free over $k[t]$ (or polynomial over the affine blowup algebras), and a torsion-free module over the principal ideal domain $k[t]$ is flat, giving flatness of $M$ over ${\mathbb{P}}^1$ (compare the chart computation of Smooth immersions, their conormal sequence, deformation charts and smooth sections). Localization for the pair $C_{X}Y\hookrightarrow M\setminus B$ gives a lift of the pulled-back cycle class to $D_i$, and two lifts differ by a class supported on $C_{X}Y$; the Cartier Gysin at infinity kills that difference because the normal line of the infinity fibre is trivial, so $j^{!}{j}_{\ast }\gamma =c_1(\mathcal{O})\cap \gamma =0$ by the basic property $c_1({\mathcal{O}}_X)=0$ of Intersection with an invertible sheaf and the first Chern class; hence ${\sigma }_i$ is well defined on Chow classes. On an integral cycle $[V]$ the closure of $V\times {\mathbb{A}}^1$ in the $t$-chart is $\mathrm{Spec}{\mathcal{O}}_V[t,I{\mathcal{O}}_V\mathrm{/}t]$, and cutting by $t$ produces ${\mathrm{gr}}_{I{\mathcal{O}}_V}{\mathcal{O}}_V$, which is the fundamental cycle of the normal cone $C_{X\cap V}V$ with its generic lengths; for a regular sequence generating $I$, Associated graded algebra of an ideal generated by a regular sequence identifies ${\mathrm{gr}}_{I}A$ with ${\mathrm{Sym}}_{A\mathrm{/}I}(I\mathrm{/}{I}^2)$, so the cone is the rank-$d$ normal bundle of Smooth immersions, their conormal sequence, deformation charts and smooth sections.